Effects of finite system-size and finite inhomogeneity on the conductivity of broadly distributed resistor networks

Monte Carlo simulation is used to investigate the critical path calculation of the conductivity of a random resistor network that has a logarithmically broad distribution of bond conductances. It has been argued that in three dimensions the conductivity prefactor exponent y is equal to the percolation correlation length exponent ν, but past numerical computations have always found y<ν. Finite-size effects are usually blamed but have never been documented. Our analysis of Monte Carlo data also finds y<ν, but we show that the result is not due to finite-size effects. Instead, the observed y<ν is due to the effects of finite inhomogeneity. The conductivity is controlled by critical conductors, but the distance between current carrying pathways is less than presumed in the theoretical arguments that lead to y=ν. The shorter separation distance results in y<ν